We have already mentioned works related
to the synthesis problem. We now give references to results on
computability and continuity. The notion of continuity with regards to Cantor
distance is not new, and for rational functions over finite alphabets, it was
already known to be decidable [1]}.
The approach of Prieur is to reduce continuity to functionality by defining from a transducer \(T\)  a transducer realizing its topological closure. We were able to extend this approach to almost all the cases we considered, except for transducers over \((\mathbb {N},\left\lbrace  < \right\rbrace )\)  with guessing allowed, so we chose a different proof strategy.
The connection between continuity and computability for functions of \(\omega \) -words over a finite alphabet has
recently been investigated in [2]} for
one-way and two-way transducers. Our results lift the case of one-way
transducers from [2]} to data words. Our
results were partially published in conference
proceedings [4]}. In that
publication, only the case of data sets equipped with the equality
predicate was considered. We now consider oligomorphic data sets
(which generalise the latter case), the data set \((\mathbb {N},\lbrace <\rbrace )\)  and
new computability notions. Despite the fact that our
results are more general, this generalisation also allows to extract
the essential arguments needed to prove this kind of
results. Moreover, compared to [4]},
we add here the possibility for the register transducer to make
non-deterministic register assignment (data guessing), which strictly
increases their expressive power.
